25-11 Linear Algebra Analytical Geometry PDF

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MEHRAN UNIVERSITY OF ENGINEERING AND TECHNOLOGY, JAMSHORO. SECOND TERM FIRST YEAR (2ND TERM) B.E.( EL, ES, TL, CS, SW, CH, PG & MN) REGULAR EXAMINATION 2011 OF 11-BATCH. LINEAR ALGEBRA AND ANALYTICAL GEOMETRY Dated: 25-11-2011.

Time Allowed: 03 Hours.

Max.Marks 80

NOTE: ATTEMPT ANY FIVE QUESTIONS. ALL QUESTIONS CARRY EQUAL MARKS.

Q.No. 01 (a) Specify the names of the following matrices.  3 0 0   (i)  0 3 0   0 0 3 

 6 5 1   (ii)  0 3 4  0 0 9

(iii)  1

1 1

(iv)   1 1

Under what condition multiplication of two matrices A and B is conformable for AB and BA? Define orthogonal matrix, hence verify that the matrix  cos x 0  sin x   0 1 0  is orthogonal matrix.   sin x 0 cos x 

(b) Distinguish between Echolon and reduced Echelon form of matrix with examples. Define rank of matrix. What will be the rank of identity matrix of order 4? 1 1 hence, find the rank of  3  2

3 4 8 3

1  2 3  1 1  7   4  7

02 (a) Define homogenous and non homogenous system of equations. What different methods you have learnt to solve system of non-homogenous linear equations? Planning and development department of Mehran University of Engineering and Technology plans to under take three projects for building classes in CE,EL and ME respectively, the list material requirements for each class in each project as follows Project1 Project2 Project3 Paint (in 10 gallons) 1 1 1 Bricks (in thousands) 1 2 3 Labour (in 100 hours) 1 4 9 If the supplier delivers 30 gallons of paint,4000 bricks and 600 hours of labour, find the number of classes built in CE,EL and ME departments (Use Gauss Elimination or Gauss Jordan method) (b) Define trival and non trival solutions for AX=0.Examine the non trival solutions for x –y +2z + w = 0 , 3x +2y + w = 0 and 4x + y +2z +2w = 0.Aso find the solution of the given system. 03 (a) How to check consistency and inconsistency for AX = b?.Under what conditions the system AX = b will possess the different solutions For what value of b and c, the system 2x +3y +5z = 9 , 7x + 3y -2z = 8 and 2x +3y +bz = c will have (i) unique solution (ii) no solution (b) Define linear dependent and linear independent vectors. Show that the unit vectors in R3 are linear independent. Also find the value of p, so that the vectors ( 1 , -1 ,p-1) , (2 ,p ,-4 ),(0 ,2+p ,-8)in R3 are linearly dependent.

Cont’d on P/-2…. (-2-) 04 (a) Write any five important properties of determinant with examples. Without expanding Prove that p

q

q p r s s r

r

s

r s  p2  q2  r2  s 2 p q



q



2

p

(b) Is it possible to find inverse of every square matrix? justify your answer with example What will be relation between two matrices A and B of order 3, if AB = BA = I3?.Use row operation or adjoint method to find inverse of   1 2  3 2 1 0   . Also verify the result.  4  2 5 

05 (a) How to locate the points in space? Show that the points (3,-1,3), (1,-1,2), (2,1,0) and (0,2,0) are the vertices of a rectangular or the vertices of a regular tetrahedron. Also find the measure of the angle between the straight lines. L:

x  2 y  3 Z 1   1 1 2

M:

x  2 y  3 Z 1   1 1 3

(b) State and prove formula for “distance of a point from a line” Find the equation of the line L through the point (5, 7/2, 5) and intersecting at right angles the line M with parametric equations. x=4+3t; y=1+t; z=-3t 06 (a) Prove the plane through a given point A (x1, yl, z1) and with non zero normal vector n=[a, b, c] has equation a (x-x1) +b (y-y1) +c(z-z1)= 0. Also verify that the straight lines x 1 y 2 z  3 x 2 y  3 z  4     and are coplanar. 2 3 4 3 4 5

(b) Rectangular coordinate system express in cylindrical and spherical coordinate systems. Transform the equation x2+y2-z2+xyz=9 into cylindrical and spherical coordinates. 07 (a) Describe Algorithm to find direction of Qibla from any part of the world, hence Find the direction of Qibla from Islamabad with latitude 33.6660N and longitude 73.1330E. (b) Give brief introduction of Multiple Integrals and its importance in practical 3

3

y

x x problems, hence Evaluate (i)  2 xy(1  x  y) dxdy (ii) e dydx 0 1 0 0

08.

DO AS DIRECTED (ANY THREE) (i) Does commutative law of multiplication holds in matrix algebra? What will be the period of matrix A, if A3 = A and A4 = A. Prove that  1  2  6 is  3 2 9  Periodic matrix, also find its period.  2

(ii)

 3 

Define minors and cofactors of any square matrix with examples. Find the value of k such that the matrix

(iii)

0

8 5   2

0 k 1

9   3   5 

is singular.

Find the equation of the plane that passes through the line of intersection of two planes 2x – y +3z= 0 and x +2y -2z – 3 =0 and perpendicular to xy plane.

(iv)

What will be the centre and radius of sphere x2 + y2 + z2 -6x + 4z = 0 . Also write (x+y)2 – z2 +4 = 0 into spherical coordinates. -------------THE END--------------...